Warps, grids and curvature in triple vector bundles

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle (Formula presented.) of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle (Formula presented.) for a vector bundle A using a connection in A to define a grid in (Formula presented.). In this case the curvature emerges from the warp theorem

Triple vector bundles in Differential Geometry

The triple tangent bundle T3M of a manifold M is a prime example of a triple vector bundle. The definition of a general triple vector bundle is a cube of vector bundles that commute in the strict categorical sense. We investigate the intrinsic features of such cubical structures, introducing systematic notation, and further studying linear double sections; a generalization of sections of vector bundles. A set of three linear double sections on a triple vector bundle E yields a total of six different routes from the base manifold M of E to the total space E. The underlying commutativity of the vector bundle structures of E leads to the concepts of warp and ultrawarp, concepts that measure the noncommutativity of the six routes. The main theorem shows that despite this noncommutativity, there is a strong relation between the ultrawarps. The methods developed to prove the theorem rely heavily on the analysis of the core double vector bundles and of the ultracore vector bundle of E. This theorem provides a conceptual proof of the Jacobi identity, and a new interpretation of the curvature of a connection on a vector bundle A

Nature, extent and ecological implications of night-time light from road vehicles

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this record1.The erosion of night‐time by the introduction of artificial lighting constitutes a profound pressure on the natural environment. It has altered what had for millennia been reliable signals from natural light cycles used for regulating a host of biological processes, with impacts ranging from changes in gene expression to ecosystem processes. 2.Studies of these impacts have focused almost exclusively on those resulting from stationary sources of light emissions, and particularly streetlights. However, mobile sources, especially road vehicle headlights, contribute substantial additional emissions. 3.The ecological impacts of light emissions from vehicle headlights are likely to be especially high because these are (i) focused so as to light roadsides at higher intensities than commonly experienced from other sources, and well above activation thresholds for many biological processes; (ii) projected largely in a horizontal plane and thus can carry over long distances; (iii) introduced into much larger areas of the landscape than experience street lighting; (iv) typically broad ‘white’ spectrum, which substantially overlaps the action spectra of many biological processes; and (v) often experienced at roadsides as series of pulses of light (produced by passage of vehicles), a dynamic known to have major biological impacts. 4.The ecological impacts of road vehicle headlights will markedly increase with projected global growth in numbers of vehicles and the road network, increasing the local severity of emissions (because vehicle numbers are increasing faster than growth in the road network) and introducing emissions into areas from which they were previously absent. The effects will be further exacerbated by technological developments that are increasing the intensity of headlight emissions and the amounts of blue light in emission spectra. 5.Synthesis and applications. Emissions from vehicle headlights need to be considered as a major, and growing, source of ecological impacts of artificial night‐time lighting. It will be a significant challenge to minimize these impacts whilst balancing drivers’ needs at night and avoiding risk and discomfort for other road users. Nonetheless, there is potential to identify solutions to these conflicts, both through the design of headlights and that of roads.The research leading to this article has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 268504 and Natural Environment Research Council grants NE/N001672/1 and NE/P01156X/1